It is shown that the presently available statistical tests for the Rasch model are insensitive to violation of the unidimensionality axiom. Two new test statistics are presented. The first one, Q1, is sensitive to the same effects as the presently available statistics, but has some desirable properties of a nonstatistical nature. The second statistic, Q2, is sensitive to violation of local stochastic independence and unidimensionality and thus fills an existing gap.

Formulas for the standard error of measurement of three measures of change—simple difference scores, residualized difference scores, and the measure introduced by Tucker, Damarin, and Messick—are derived. Equating these formulas by pairs yields additional explicit formulas which provide a practical guide for determining the relative error of the three measures in any pretest-posttest design. The functional relationship between the standard error of measurement and the correlation between pretest and posttest observed scores remains essentially the same for each of the three measures despite variations in other test parameters (reliability coefficients, standard deviations), even when pretest and posttest errors of measurement are correlated.

A unidimensional latent trait model for responses scored in two or more ordered categories is developed. This “Partial Credit” model is a member of the family of latent trait models which share the property of parameter separability and so permit “specifically objective” comparisons of persons and items. The model can be viewed as an extension of Andrich's Rating Scale model to situations in which ordered response alternatives are free to vary in number and structure from item to item. The difference between the parameters in this model and the “category boundaries” in Samejima's Graded Response model is demonstrated. An unconditional maximum likelihood procedure for estimating the model parameters is developed.

Two algorithms are described for marginal maximum likelihood estimation for the one-parameter logistic model. The more efficient of the two algorithms is extended to estimation for the linear logistic model. Numerical examples of both procedures are presented.

One of the intriguing questions of factor analysis is the extent to which one can reduce the rank of a symmetric matrix by only changing its diagonal entries. We show in this paper that the set of matrices, which can be reduced to rank r, has positive (Lebesgue) measure if and only if r is greater or equal to the Ledermann bound. In other words the Ledermann bound is shown to be almost surely the greatest lower bound to a reduced rank of the sample covariance matrix. Afterwards an asymptotic sampling theory of so-called minimum trace factor analysis (MTFA) is proposed. The theory is based on continuous and differential properties of functions involved in the MTFA. Convex analysis techniques are utilized to obtain conditions for differentiability of these functions.

Restricted multidimensional scaling models [Bentler & Weeks, 1978] allowing constraints on parameters, are extended to the case of asymmetric data. Separate functions are used to model the symmetric and antisymmetric parts of the data. The approach is also extended to the case in which data are presumed to be linearly related to squared distances. Examples of several models are provided, using journal citation data. Possible extensions of the models are considered.

Three alternative estimation procedures for factor analysis based on the instrumental variables method are presented. These procedures are justified by the method of least squares. Formulas for asymptotic standard errors of factor loadings are derived. The procedures are empirically compared to the method of maximum likelihood. The conclusion, based on the data used in this study, is that two of the procedures seem to work well.